Antenna array synthesis method



Aug- 6, 1963 c. J. sLETTEN 3,100,300

ANTENNA ARRAY SYNTHESIS METHOD Aug. 6, 1963 Filed oct. io, 195e C. J. SLETTEN ANTENNA ARRAY SYNTHESIS METHOD 2 Sheets-Sheet 2 hired Stat-es The invention described herein may be manufactured and used by or for the United States Government for governmental ypurposes without payment to me of any royal-ty thereon.

This invention relates :generally to antennas and more particularly to a method of synthesizing linear antenna arrays by utilizing the natural phase distribution that exists on a transmission line. 'Ihe far-fie1d pattern function obtained with equispaced elements is decomposed in terms of the feeding coeilicients realizable along the feeding guide. Such functions are suciently general to permit synthesis of any arbitrary pattern with `arbitrary phase.

In utilizing the method of synthesis of this invention, use is madeof the natural phases which occur on a feeding waveguide or transmission line to feed a linear array of any kind; the radiators are spaced about g/ 4 apart and produce complex antenna patterns. Only amplitude, rather than both amplitude and phase, is used to control the radiated power.

Prior techniques have been more complicated and difficult `to use because they 'involve .approximating a continuous aperture distribution by means of `discrete radiating elements, and, since phasing is approximately obtained by variable longitudinal -spacing between elements, pattern calculation is dicult.

Accordingly, it is -an object of this invention to provide a technique or method of antenna synthesis which produces more precise control of antenna beam shapes.

Another object of the invention involves the simplication of -antenna designs for complex beam shapes such as are used for Ground Control Approach (GCA) search :and mapping.

Still another object `of the invention is the utilization of the natural phases on a 'feeding waveguide or transmission line to produce any desired coverage pattern.

A further object of the invention involves the production of a new antenna device based on a new method of array synthesis.

A still further object of .the invention involves an antenna device wherein the relative electrical 4phase of a radiator is `obtained from a constant longitudinal spacing together with the position of the element Awith respect to the center eline of the antenna.

These and other advantages, features `and objects of the invention will become more apparent .from the following description taken in connection 'with the illustrative embodiments in the yaccompanying drawings, wherein:

FIGURE l is an illustration representing-a linear array; FIGURE 2 depicts a microwave antenna suitable for use in the resolution of a radiated pattern in terms of phase on a transmission line;

FIGURE 3 illustrates an ideal yfield pattern;

FIGURE 4 shows -the pattern Aof FIGURE 3 broken into-odd and even parts;

5 shows -a seven slot array Ydesigned in accordance with the method of this invention; and

`FIGURE 6 represents an I-I-plane pattern of the array of FIGURE 5.

-The mathematical basis of this invention is treated as a problemY in one dimension, the -0 plane, lalthough arent 3,100,300 Patented Aug. 6, `1l-063 P ICC with rectangular y'apertures the problem can often be `sepanated into .two one-dimensional problems.

FIGURE l shows 0 measured from the normal to the array. The radiation pattern produced by a l-inear array 0f omnidirectional elements will always have even symmetry `with respect to the line of the array unless the elements are made directive by 'a ground screen or some other device.

The required power pattern E(0)`E*(0) is specilied in the interval --`90090. It is assumed that individual radiating elements produce identical radiation f(0)=D(0) 2 Rue frt=N where Rn is the amplitude of excitation, on is the phase of excitation, )t is the wavelength, d is the element spacing, and :,l/ is equal to sin 0.

FIGURE 2 illustrates how f(0) lcan be resolved into component functions that are simply related to phase on the feeding line. On a transmissiondine shunt loaded with real ladmittances at nodes or anti-nodes in the standing-wave pattern, the phase increases from the feeding end at each successive maximum and minimum of the transmission-line iields. When 180 phase reversal of the radiator is possible, such as is obtained by crossing the leads on `a dipole or, as show-n here, by placing longitudinal shunt slots on opposite sides of the center lline on the broad Aface of -a rectangular waveguide in the TEM, mode, the values of e1"ibn can be 1, -1, i, and

The resolu-tion of a radiated pattern of each yelemental array of slots in FIGURE 2 in terms `of phase on a 'transmission line may be expressed by:

where each term in each of the functions corresponds to a particular radiator and each slot in FIGURE 2 -is labeled with the letter of its function; the terms to the left corresponding to Ithe radiators shown at the left, etc.

where ,-i-j is 90 lead in time; and (N +1) is the total number rof elements left or right of center of phase.

InvFIGUREV 2' the phase reference is the slot in the Y center of the array. The -feeding coeilicients Am, Bn, Cm and Dn are assumed to be real, and are adjustable by the coupling conductance g of the radiator. The

phase'of the element is obtained from the longitudinal be decomposed into even and odd parts since electromagnetic ieldsare linear and can be superposed on one another. Hence,

The degree of approximation to f(6) 'will of course 'depend on the number of terms -in the series, that is,

on the number of radiators used. The coefficients An, Bn, Cn, and Dn, can be determined in any manner; however, the Fourier approximation is signicantly easier to compute since each added temi i-s independent of earlier terms. rllhus,

D..=1f zff,. 0 sin @fidi To check orthogonal-ity, :it is noted that Sill (mi-V2M# or GOS (MH/2)@ It can be mathematically proved Ithat this decomposition is suflicient to describe any arbitrary pattern f().

1*(0) lras been developed in a least square approximation in 1p=sin 6. a least square approximation in 6 is wanted a system of equations mus-t be evaluated Where integrals like are typical. 'Ilhese can usually be expressed in Bessel or Struve functions when K0) is simple.

Only when 1:1 or d=M2 will the expansion interval agree with the physical interval 090". When l 1, d ?t/2, part of f(0) will not be visible. When l 1, d )\/2, K9) will start repeating lwhen 0=sin1l.

Experimental eiorts in the application of the method of the invention to the double-step -function can be illustrated by the representations of the ideal function shown in FIGURE 3. Both waveguide and two-Wire line-iceding systems have been used, While the double-step function was selected because it minimized lthe number of spaced elements and their resulting mutual couplings.

The pattern in FIGURE 3 can tbe thought of as made up of a cosine symmetric main lobe and `a step function on one side of lthe 0=0 axis. To simplify the design w(0) produces the main lobe and a combination of fv(0) and fu(9) produces the step function. Both odd and even patterns are Athereby obtained in phase quadrature where they do not destructively interfere with each other.

As 'a rst step, suppose fv produces an odd function as shown in dotted lines in FIGURE 4, the even function being displayed in `solid lines. Tlhen 0 l BFL-a sin (warg-bdw a sin (Magda Note that when inw) generates a constant function equal to 1/2, the series reduces to one term, C0=1/2.

fw represents the peak of the pattern and can be contputed according to Dolph-Tchebycheft theory of linear arrays to give ma) :3.259 cos 90 k+2.170

cos 270 ktm-.868 cos 450 kv# which represents the formula with coeicients normalized to present the peak 16 db above the step function.

Now, to produce a step function of .unity amplitude in the interv-al between 090 and approximately zero in the interval O-S", one can write for the first seven radiators:

fstep=fv+fu=05+0536 sin 90 [tgl/+0213 sin 270 kil/+0128 Sin 450 loll Where k=)\g/A. The total pattern equals the sum of fu,

and fw. 'The iirst slots on each side of the phase reference center are spaced Apg/4, and all other slots are spaced Ag/Z. An illustration of an array with a feeding input or bolometer F :and matched load L for producing such a single-step pattern -is shown in FIGURE 5; pattern results are shown in FIGURE 6. The slots of the embodiment are oriented in the same direction to achieve the same polarization. By placing the center slot on the opposite side of the center line the pattern could be reversed. In FIGURE 5, slots S1, S2, and S3 are at one quarter wavelength spacing while radiators S4, S5, S6, and S7 yare `at half wavelength spacing. v

Ithe slot coupling to the waveguide is controlled by transverse spacing. The couplings and transverse spacings can be directly calculated once the relative radiated 'power at each longitudinal point is known. The field (E) of each elemental slot is combined at radiating points power (P). The combined pattern distribution, which J results from the pattern distribution lof `each yor" the elements, produces the total eld of the nal array.

Thus, it can be seen that by assuming a desired pattern, dividing it into odd and even functions, determining the relative radiating power of the elements, combining all elements of the same longitudinal spacing in accordance with the nal power distribution, and choosing elements coupling to the transmission line according to the calculated power distribution; an antenna design will result with at least one radiator having approximately one quarter Wavelength spacing. The invention is applicable to any array of radiating elements where there is 180 phase reversal with respect to a feeding transmission line, for example, With a balanced two-wire line and dipoles connected thereto.

Although the invention has been described with reference to a particular embodiment, it will be understood to those skilled in the art that the invention is capable of a Variety of alternative embodiments Within the spirit and scope of the appended claim.

I claim:

A .method of controlling the radiation pattern established by an antenna array which comprises the steps of choosing a radiation pattern corresponding to a Fourier series of odd and even functions, and causing radiation to occur along said array at positions such that the energy amplitude :at each position conforms to an element of said Fourier series in such manner as to provide a pattern summation corresponding to the quarter Wavelength summation value of said series.

References Cited in the tile of this patent UNITED STATES PATENTS 2,541,910 Bangert et al Feb. 13, 1951 2,639,383 Gruenberg May 19, 1953 2,648,839 Ford et al Aug. 11, 1953 2,659,005 Gruenberg Nov. 10, 1953 2,679,590 Riblet May 25, 1954 2,712,604 Thomas et al. July 5, 1955 2,756,421 Harvey et al. July 24, 1956 2,807,018 Woodward Sept. 17, 1957 2,840,818 Reed et al June 24, 1958 2,854,666 Gamer-tsfelder Sept. 30, 1958 2,874,382 Stavis Feb. 17, 1959 FOREIGN PATENTS 1,014,859 France June 25, 1952 725,386 Great Britain Mar. 2, 1955 OTHER REFERENCES Antennas by Kraus, copyright 1950, pages 97 to 109. 

